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Lorentz's reciprocity lemma and Feld-Tai reciprocity theorem show the effectof interchanging the action and reaction in Maxwell's equations. We derive afree-space version of these reciprocity relations which generalizes theconservation of the momentum-energy tensor. This relation corresponds to theinterference conservation of electromagnetic waves. We show that for anytransformation or symmetry that leaves Maxwell's equations invariant, we canmodify the reciprocity relation to introduce a conserving density, optical fluxand stress tensor extending Noether's theorem to a different context. We applythis method to transformations that can be expressed as Hermitian operators andmore specifically we define the operators associated with the optical energy,spin, linear and angular momentum.

Spin and angular momentum operators and their conservation